Thanks for those that have responded. We are 99.9% sure that it is engine oil and not transmission fluid that is getting into the cooling system. The level of transmission fluid has remained constant while the engine oil level has fallen.
All of the hoses were replaced and the radiator was sent to a radiator shop to be flushed, so hopefully we are good on that front.
The first time that the problem occured the level of engine oil went down relatively fast (5 quarts in about about 10 miles) while the second time it went doen about 1 quart in 100 miles. Could this be telling us something?